一类非线性双曲型方程的Galerkin方法

Galerkin Method for a Nonlinear Hyperbolic Equation

主要讨论了平面有界凸多角形区域上的一类非线性双曲型方程utt- .( a( x,u) u) =f ( x,u)u( x,0 ) =u0 ( x)ut( x,0 ) =φ( x)u( x,t) =0 ( x,t)∈Ω× [0 ,T]x∈Ωx∈Ω( x,t)∈ Ω× [0 ,T]的 Galerkin有限元方法 ,首先给出了所讨论问题的 Galerkin有限元方法的离散格式 ,其次对所讨论问题的解与其离散问题的解之间的误差进行了分析研究 ,最后利用椭圆投影算子的性质 ,得到了 L2 模和能量模方面的一些误差估计。

Galerkin finite element methods for an initial boundary value problem of a nonlinear second order hyperbolic equation in two dimension space are treated. The problemu tt -·(a(x,u)u)=f(x,u) u(x,0)=u 0(x) u t(x,0)=φ(x) u(x,t)=0 (x,t)∈Ω×[0,T] x∈Ω x∈Ω (x,t)∈Ω×[0,T]Firstly, this paper studies the discrete approximation scheme of the Galerkin finite element methods for the discussed problem. Secondly, the error estimates for the solution of the discussed problem and the solution of the discrete approximation scheme are considered. Finally, by means of the elliptic projection operator, some error estimates on norm and energy norm are derived.